Questions tagged [regularity]

regularity of solutions of PDEs.

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Different ways to prove $L^p$-estimates for the heat equation

Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$. $$ \begin{cases} \partial_t u = \Delta u + f \\ u(0) = 0. \end{cases} $$ It is well-...
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4 votes
1 answer
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Reference to a Classical Regularity Theorem

(Edited) I need a reference to the following result: If $u \in H^2(B_1^+) \cap {\rm Lip}(B_1^+)$ satisfies \begin{cases} {\rm div}(F(x,u,\nabla u)) = F_0(x,u,\nabla u) \quad & {\rm in} \ B_1^+ ...
artful_dodger's user avatar
3 votes
2 answers
251 views

Regularity of a conformal map

Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (...
AmorFati's user avatar
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2 votes
0 answers
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Reference request : Global boundedness of weak solution for Neumann problem

I have some question on global boundedness of weak solution to Neumann problems. Let $u\in W^{1,2}(\Omega)$ is a weak solution for Neumannn problem $$ \mathrm{div} (A \nabla u )= \mathrm{div}\, F\quad ...
Will Kwon's user avatar
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3 votes
0 answers
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$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation

Consider a Fokker-Planck equation: $$ \partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0, $$ with initial condition ...
PPDE's user avatar
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1 vote
0 answers
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Regularity theory for parabolic PDEs in fractional Sobolev spaces

I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
George's user avatar
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2 votes
0 answers
72 views

wave equation with non-smooth coefficients

Let us consider the equation $$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$ subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
Ali's user avatar
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2 votes
1 answer
357 views

Regularity on the boundary for the heat equation with linear source

This is probably a known problem but I was not able to find exactly what I am looking for. I have the following linear heat equation with zero-flux boundary conditions: \begin{equation} \begin{cases}...
MathMax's user avatar
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3 votes
1 answer
232 views

Regularity and normal trace of "Hdiv" measures

In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$. I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
leo monsaingeon's user avatar
3 votes
1 answer
228 views

Inequality for initial data

I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
Sigma's user avatar
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2 votes
0 answers
194 views

$W^{2,p}$-estimates for Neumann boundary condition to Poisson equation

Consider the following Poisson-Neumann problem in a lipschitz bounded domain $\Omega\subset \mathbb{R}^3$: $-\Delta u=F,\quad \partial_n u\restriction_{\partial\Omega}=0$. Here $F\in L^p(\Omega)$. ...
FluidFlow's user avatar
1 vote
0 answers
63 views

Axis regularity in cylindrical coordinates: conditions on the non linear terms?

I've been working on this for months and can't find a good answer. I'm looking at the incompressible Euler equations in cylindrical coordinates ($r$, $\theta$, $z$), and I am looking at the non ...
Meuoua's user avatar
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3 votes
1 answer
408 views

regularity of p-harmonic functions

We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$. Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
user143090's user avatar
1 vote
0 answers
33 views

free boundary of a p-harmonic function

let $u$ be a p-harmonic function in $\Omega \subset \mathbb R^N$. We already know that the set $\{u=0\}$ is locally a $C^{1,\alpha}$ hypersurface at the points where $\nabla u\neq 0$. What can be ...
Harish's user avatar
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2 votes
0 answers
684 views

Regularity of locally finite Borel measure

Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...
Michael's user avatar
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0 votes
1 answer
260 views

Estimate for Laplace equation with Neumann boundary on manifold with corner

Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$. ...
DLIN's user avatar
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1 vote
0 answers
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Integrability condition on function determining PDE domain

I'm currently looking through the following paper which examines some dynamics of the Airy$_2$ process: https://arxiv.org/pdf/1106.2717.pdf On page 2, there appears a PDE of the form $\partial_t u +...
Marc Vaisband's user avatar
2 votes
0 answers
58 views

Time derivative in parabolic Hölder spaces

Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$. Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
foo90's user avatar
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5 votes
1 answer
661 views

Embedding of weighted sobolev space with exponential weights

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. ...
John's user avatar
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2 votes
0 answers
89 views

A question about how to use the convexity condition?

At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say that since $p$ is convex, we can deduce that $$ \...
Hheepp's user avatar
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8 votes
0 answers
439 views

Regularity result for the boundary value problem for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider the following boundary value problem for the heat equation: ...
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3 votes
1 answer
322 views

Reference request: Schauder estimates for parabolic equations

Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)? Any reference would be highly appreciated.
Math_tourist's user avatar
2 votes
0 answers
78 views

Wave equation regularity

I have an equation of the type $$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$ for $x\in (0,1)$ and $t>0$, where $\hat{\rho}$ and $\hat{E}$ are constants, $u(0,t)=u(1,t)=0$, $u(x,0)=p(x)$, $u_t(x,0)=q(...
REVB's user avatar
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5 votes
1 answer
377 views

Oscillation and Hölder continuity

Where can I find a proof of the following fact? If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda ...
user avatar
1 vote
0 answers
110 views

Does the regularity of the initial data have to agree with the solution's spatial regularity in evolutionary PDEs?

Let's say we have some evolutionary PDE and the initial data $u_0$ is in the space $X$. For example $X=H^s(\Omega)$ for some $s$. My question is if the solution has to have the same spatial regularity,...
Cahn's user avatar
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6 votes
1 answer
377 views

Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?

$\newcommand{\R}{\mathbb R}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\de}{\delta}$ $\newcommand{\sig}{\sigma}$ $\newcommand{\Average}[1]{\left\langle#1\right\rangle} $ $\newcommand{\IP}[2]{\Average{...
Asaf Shachar's user avatar
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0 votes
1 answer
91 views

$2n$-regular graphs with maximal chromatic number

Let $n\geq 1$ be an integer. Suppose $m\geq 2n+1$ is an integer. We construct the graph $\mathbb{Z}_m = (\mathbb{Z}/m\mathbb{Z}, E_m)$ where $$E_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ ...
Dominic van der Zypen's user avatar
0 votes
0 answers
118 views

Biharmonic equation

Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation \begin{equation}\label{\star} \Delta^2u+V(x)u=g(x, u)+K(x)u, \end{equation} where $|g(x,s)|\leq \varepsilon|...
Pádua's user avatar
  • 67
5 votes
1 answer
318 views

Regularity of harmonic forms on manifolds-with-boundaries

Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
Siran Victor Li's user avatar
2 votes
1 answer
173 views

Well-posedness of wave equations with time-dependent coefficient

Let us consider the following wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \\ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\ y(0,x)...
Gustave's user avatar
  • 545
2 votes
1 answer
455 views

Bounded solution for parabolic equation

Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\...
S. Maths's user avatar
  • 561
10 votes
4 answers
2k views

Elliptic regularity on compact manifold without boundary

Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this: For any $u\in H^1(M)$, ...
S. Maths's user avatar
  • 561
9 votes
1 answer
1k views

Global regularity for Neumann problem

Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
Guy Fsone's user avatar
  • 1,033
4 votes
1 answer
838 views

Evans-Krylov theorem

Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
user128943's user avatar
1 vote
1 answer
270 views

Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$. Explicitly, let $\M$ be a smooth oriented Riemannian manifold. ...
Asaf Shachar's user avatar
  • 6,611
3 votes
2 answers
229 views

Density on Hölder spaces whose elements vanish on the boundary

I would like to ask the following problem. Let $\Omega$ be a $C^{r+1,\alpha}$ domain, $r\in \mathbb{N}, 0<\alpha<1.$ We denote $$C^{r,\alpha}_{0}(\overline{\Omega})=\{f\in C^{r,\alpha}(\...
Hoàng Phương Nguyễn's user avatar
1 vote
0 answers
95 views

2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of $$ u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\ u(0) = u_0 \text{ in } H, u'(0)...
sgr's user avatar
  • 65
7 votes
1 answer
206 views

Is a Sobolev map with smooth minors smooth on the whole domain?

Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$. ...
Asaf Shachar's user avatar
  • 6,611
0 votes
1 answer
388 views

Regularity of Laplace equation with Dirichlet data on a part of the boundary

From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system \begin{align} -\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\ u &= g &\text{on}\ \...
Julienne Franz's user avatar
0 votes
1 answer
489 views

$H^2$ regularity for Laplace equation with Robin-Robin boundary condition

From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP \begin{align} -\Delta u &= 0 & \text{in}\ \Omega\\ -\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\ \end{align} where $a&...
Julienne Franz's user avatar
3 votes
1 answer
759 views

Higher regularity of solutions for Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an open bounded Lipschitz domain of class $C^{1,1}$ with boundary $\partial \Omega = \Gamma_i \cup \Gamma_o$, $\Gamma_i \cap \Gamma_o = \emptyset$ and dist$(\...
Julienne Franz's user avatar
6 votes
0 answers
254 views

Is a Sobolev map with invertible smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
Asaf Shachar's user avatar
  • 6,611
3 votes
2 answers
1k views

Reference for De Giorgi-Nash-Moser theory

I am interested in Holder regularity for equations of the form $$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic. This was proved in the seminal paper of John Nash ...
Adi's user avatar
  • 483
2 votes
1 answer
276 views

Is this approach for establishing regularity of harmonic maps between manifolds valid?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...
Asaf Shachar's user avatar
  • 6,611
1 vote
0 answers
43 views

Hidden regularity for the coupled wave equation with dynamaic boundary condition

We have the equation \begin{equation} \left\{ \begin{array}{rrrr} u_{tt}-\Delta u=0,&\text{in} & \Omega \times ]0,T[ & \left( 1.1\right) \\ u=0, & \text{on } & \Gamma _{0}\...
Gustave's user avatar
  • 545
0 votes
0 answers
74 views

Looser condition for regularity for Neumann problems

If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then $$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$ Let's say we are studying rotationally symmetric solutions to ...
Tim Carson's user avatar
5 votes
1 answer
425 views

Dependence of the Hölder exponent in De Giorgi-Nash-Moser

I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity. More precisely: suppose $u$ satisfies weakly $$ D_i(a^{ij}D_ju)=f $$ on the $d$-...
m7e's user avatar
  • 458
1 vote
0 answers
46 views

Harnack type Estimates for a p-Poisson equation with constant source term

Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE $$ -\Delta_p u=1\,\,\mbox{in $B$} $$ Let another function $f$ be such that $$ \begin{cases} -\Delta_p f =1 \;\;\mbox{in $B$}\\ f=0 \...
Harish's user avatar
  • 251
3 votes
0 answers
99 views

Is there a weak isometric completion to a $W^{2,2}$ isometric immersion?

Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$. Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$). Suppose we are given a $W^{2,2}$ isometric immersion $...
Asaf Shachar's user avatar
  • 6,611
1 vote
1 answer
185 views

Is there any "extra regularity" to the solution to Poisson's equation posed on a 3-dimensional polyhedron?

I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form $$ \|u\|_{H^{1+\epsilon}(\Omega)} \le C \|f\|_{L^2(\Omega)} $$ for some $\epsilon >0 $ ...
fred's user avatar
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